IMRN International Mathematics Research Notices
2002, No. 20
A Counterexample to the Hodge Conjecture
Extended to Kähler Varieties
Claire Voisin
1
Introduction
If X is a smooth projective variety, it is in particular a Kähler variety, and its cohomology
groups carry the Hodge decomposition
Hk (X, C) = ⊕p+q=k Hp,q (X).
(1.1)
A class α ∈ H2p (X, Q) is said to be a rational Hodge class if its image in H2p (X, C)
belongs to Hp,p (X). As is well known, the classes which are Poincaré dual to irreducible
algebraic subvarieties of codimension p of X are Hodge classes of degree 2p. The Hodge
conjecture asserts that any rational Hodge class is a combination with rational coefficients of such classes.
In the case of a general compact Kähler variety X, the conjecture above, where
the algebraic subvarieties are replaced with closed analytic subsets, is known to be
false (cf. [13]). The simplest example is provided by a complex torus endowed with
a holomorphic line bundle of indefinite curvature. If the torus is chosen general enough,
it will not contain any analytic hypersurface, while the first Chern class of the line
bundle will provide a Hodge class of degree 2.
In fact, another general method to construct Hodge classes is to consider the
Chern classes of holomorphic vector bundles. In the projective case, the set of classes
generated this way is the same as the set generated by classes of subvarieties. To see
this, one introduces a still more general set of classes, which is the set generated by the
Received 22 November 2001. Revision received 21 January 2002.
1058 Claire Voisin
Chern classes of coherent sheaves on X. Since any coherent sheaf has a finite resolution
by locally free sheaves, it follows from Whitney’s formula that the set generated by
Chern classes of coherent sheaves is not larger than the set generated by Chern classes
of locally free sheaves. On the other hand, this latter set obviously contains the classes
of subvarieties (we compute for this the Chern class of degree 2p of IZ for Z irreducible
of codimension p, and we show that it is proportional to the class of Z). Finally, to
see that the Chern classes of locally free coherent sheaves can be generated by classes
of subvarieties, we first reduce, twisting by a very ample line bundle, to the case of
globally generated locally free sheaves E. Such an E is the pullback of the tautological
quotient bundle Q on a Grassmannian G by a morphism from X to G. Then we use the
fact that the Chern classes of Q are generated by classes of subvarieties, because the
whole cohomology of the Grassmannian is generated by such classes.
In the general Kähler case, none of these equalities between the three sets of
Hodge classes introduced above holds. The only obvious result is that the space generated by the Chern classes of analytic coherent sheaves contains both the classes which
are Poincaré dual to irreducible closed analytic subspaces and the Chern classes of holomorphic vector bundles (or locally free analytic coherent sheaves). The example above
shows that a Hodge class of degree 2 may be the Chern class of a holomorphic line
bundle, even if X does not contain any complex analytic subset. On the other hand, we
show, in the appendix, that coherent sheaves on a compact Kähler manifold X do not
need to admit a resolution by locally free sheaves (although it is true in dimension 2,
see [8]), and that more generally, X does not necessarily carry enough vector bundles to
generate the Hodge classes of subvarieties or coherent sheaves. Hence, the set Hdg(X)an
generated by the Chern classes of analytic coherent sheaves is actually larger than the
two others.
Notice that using the Grothendieck-Riemann-Roch formula (cf. [3], extended
in [7] to the complex analytic case), we can give the following alternative description of
the set Hdg(X)an : it is generated by the classes
φ∗ ci (E),
(1.2)
where φ : Y → X is a morphism from another compact Kähler manifold, E is a holomorphic vector bundle on Y, and i is any integer. Another fact which follows from iterated
applications of the Whitney formula, is that the set which is additively generated by the
Chern classes of coherent sheaves is equal to the set which is generated as a subring of
the cohomology ring by the Chern classes of coherent sheaves. Thus this set is as big
and stable as possible.
A Counterexample to the Hodge Conjecture for Kähler Varieties 1059
Now, since we do not know other geometric ways of constructing Hodge classes,
the following seems to be a natural extension of the Hodge conjecture to Kähler varieties.
Are the rational Hodge classes of a compact Kähler variety X generated over Q
by rational Chern classes of analytic coherent sheaves on X?
Our goal in this paper is to give a negative answer to this question. We show the
following theorem.
Theorem 1.1. There exists a 4-dimensional complex torus X which possesses a nontrivial Hodge class of degree 4, such that any analytic coherent sheaf F on X satisfies
c2 (F) = 0.
In the appendix, we give a few geometric consequences of a result of Bando and Siu [1],
extending Uhlenbeck-Yau’s theorem. We show in particular that for a compact Kähler
variety X, the analytic coherent sheaves on X do not need to admit finite resolutions by
locally free coherent sheaves. This answers a question asked to us by L. Illusie and the
referee.
Notation.
In this paper, the Chern classes considered are the rational Chern classes
ci (F) ∈ H2i (X, Q),
(1.3)
for F a coherent analytic sheaf on a complex manifold X.
2
A criterion for the vanishing of Chern classes
of coherent sheaves
We consider in this section a compact Kähler variety X of dimension n ≥ 3 satisfying
the following assumptions:
(1) the Néron-Severi group NS(X) generated by the first Chern classes of holomorphic line bundles on X is equal to 0;
(2) X does not contain any proper closed analytic subset of positive dimension;
(3) for some Kähler class [ω] ∈ H2 (X, R) ∩ H1,1 (X), the set of Hodge classes
Hdg4 (X, Q) is perpendicular to [ω]n−2 for the intersection pairing
H4 (X, R) ⊗ H2n−4 (X, R) −→ R.
(2.1)
Our aim is to show the following result.
Proposition 2.1. If X is as above, any analytic coherent sheaf F on X satisfies c2 (F) = 0.
1060 Claire Voisin
Proof. As in [2], the proof is by induction on the rank k of F. We note that because
dim X ≥ 3, the torsion sheaves supported on points on X have trivial c1 and c2 . On the
other hand, assumption (2) implies that torsion sheaves are supported on points. This
deals with the case where rkF = 0. Furthermore, this shows that we can restrict to
torsion free coherent sheaves.
If F contains a nontrivial subsheaf G of rank < k, we have the exact sequence
0 −→ G −→ F −→ F/G −→ 0,
(2.2)
with rkG < k and rkF/G < k. Assumption (1) and the induction hypothesis then give
c1 (G) = c2 (G) = 0,
c1 (F/G) = c2 (F/G) = 0.
(2.3)
Therefore Whitney’s formula implies that c2 (F) = 0.
We are now reduced to study the case where F does not contain any nontrivial
proper subsheaf of smaller rank. By assumption (2), F is locally free away from finitely
many points {x1 , . . . , xN } of X. We show now that there exists a variety
τ : X̃ −→ X,
(2.4)
which is obtained from X by finitely many successive blowups with smooth centers (and
in particular is also Kähler), so that τ restricts to an isomorphism
∼ X − x1 , . . . , xN ,
X̃ − ∪i Ei =
(2.5)
where Ei := τ−1 (xi ) and such that there exists a locally free sheaf F̃ on X̃ which is
isomorphic to F on the open set X̃ − ∪i Ei : indeed, the problem is local near each xi .
Choosing a finite presentation
OlX −→ OrX −→ F −→ 0
(2.6)
of F near xi , we get a morphism to the Grassmannian of k-codimensional subspaces of
Cr , which is well defined away from xi , since F is free away from xi . Using the finite presentation, this morphism is easily seen to be meromorphic. Hence by the Hironaka desingularization theorem [5], this morphism can be extended after finitely many blowups.
(A priori, [5] works in the algebraic context, but since our meromorphic map takes value
in a projective variety and has an isolated point as indeterminacy locus, we can reduce
easily to the situation considered in [5].) Then the pullback of the tautological quotient
bundle on the Grassmannian will provide the desired extension.
A Counterexample to the Hodge Conjecture for Kähler Varieties 1061
Note that because F does not contain any nonzero subsheaf of smaller rank, the
same is true for F̃: indeed, assuming there exists a nonzero coherent subsheaf of smaller
rank G ⊂ F̃, consider
R0 τ∗ G ⊂ R0 τ∗ F̃.
(2.7)
This is a nontrivial coherent subsheaf of smaller rank. Now note that R0 τ∗ F̃ is isomorphic
to F away from the xi ’s, hence it is contained in the bidual F∗∗ of F. But the natural
inclusion
F
F∗∗
(2.8)
is an isomorphism away from the xi ’s, so that its cokernel T is a coherent sheaf supported
on the xi ’s. The sheaf
G := Ker R0 τ∗ G −→ T
(2.9)
is then nontrivial of rank equal to the rank of G, hence smaller than k, and it is contained
in F, which is a contradiction.
This fact implies that F̃ is h̃-stable for any Kähler metric h̃ on X̃. The theorem of
existence of Hermitian-Yang-Mills connections [10] then provides F̃ with a HermitianEinstein metric k̃ for any Kähler metric h̃ on X̃. This means that the curvature R̃ ∈
Γ (Hom(F̃, F̃) ⊗ Ω2X̃,R ) of the metric connection on F̃ associated to k̃ is the sum of a diag-
onal matrix with all coefficients equal to µ̃ω̃ and of a matrix R̃0 whose coefficients are
(1, 1)-forms annihilated by the Λ operator relative to the metric h̃. (The connection is
then said to be Hermitian-Yang-Mills.) Here ω̃ is the Kähler form of the metric h̃ and µ̃
is a constant coefficient, equal to
2iπ
c1 F̃ [ω̃]n−1
,
k[ω̃]n
(2.10)
where [ω̃] ∈ H2 (X̃, R) denotes the de Rham class of the form ω̃.
We assume chosen small neighbourhoods Vi of xi in X, and closed (1, 1)-forms
ωi on X which vanish outside τ−1 (Vi ), and restrict to a Kähler form on the divisor
Ei = τ−1 (xi ). That such forms exist follows from the fact that for each i, some com
bination j −nij Eij , nij > 0, of the components Eij of the divisor Ei is ample when
restricted to Ei . It then follows that there exists a Hermitian metric on the line bundle
OX̃ (− j nij Eij ) whose Chern form restricts to a Kähler form on Ei . We may obviously
1062 Claire Voisin
assume that this metric is the constant metric away from τ−1 (Vi ), where we use the
natural isomorphism
OX̃
−
∼O
=
X̃
nij Eij
j
(2.11)
away from Ei , and then take for ωi the Chern form of this metric.
Then we will choose for Kähler form ω̃ the form
∗
ωλ = τ ω + λ
ωi
(2.12)
i
which depends on λ > 0 and is easily seen to be Kähler for sufficiently small λ.
Let h̃ = hλ , k̃ = kλ , R̃ = Rλ . We denote by η0λ the closed 4-form
η0λ
= tr
R0λ
2iπ
2
.
(2.13)
We now prove that Rλ tends to 0 with λ in the L2 -sense away from the Vi ’s. The argument is an extension of Lübke’s inequality [6] which proves that a Hermitian-Yang-Mills
connection on a vector bundle E with c1 (E)[ω]n−1 = c1 (E)2 [ω]n−2 = c2 (E)[ω]n−2 = 0,
where [ω] is the class of the Kähler form on the basis, is in fact flat. We claim the following proposition.
Proposition 2.2. For any differential (2n − 4)-form α on X, the integral
X−∪i Vi
η0λ ∧ α
tends to 0 with λ.
(2.14)
Before proving this proposition, we show how it implies that c2 (F) = 0, thus completing
the induction step.
Poincaré duality will provide an isomorphism
∼ H2n−4 X, ∪i Vi , R ∗ ,
H4 X − ∪i Vi , R =
(2.15)
which is realized by integrating closed 4-forms defined over X − ∪i Vi against closed
(2n − 4)-forms vanishing on the Vi ’s. Next because dim X ≥ 3 we have the isomorphisms
∼ H4 X − ∪i Vi , R ,
H4 (X, R) =
∼ H2n−4 (X, R),
H2n−4 X, ∪i Vi , R =
(2.16)
A Counterexample to the Hodge Conjecture for Kähler Varieties 1063
which are compatible with Poincaré duality. Now the cohomology class of the closed
4-form η0λ is easily computed to be
2
0
2 µλ 2
µλ
ηλ = −2c2 F̃ + c1 F̃ −
ωλ ∪ c1 F̃ + k
ωλ .
iπ
2iπ
(2.17)
∼ X − ∪i Vi satisfies
Hence its restriction to X̃ − ∪i τ−1 (Vi ) =
0
ηλ |X−∪
i Vi
=
2
µλ
2
− 2c2 (F) + k
[ω]
2iπ
,
(2.18)
|X−∪i Vi
In order to show that c2 (F) = 0, it then suffices by (2.15), (2.16), and (2.18) to
show that for any closed (2n − 4)-form α on X, vanishing on ∪i Vi , we have
2 µλ
0
ηλ − k
ω2 ∧ α = 0.
2iπ
X
(2.19)
But this integral is independent of λ and so it suffices to show that
2 µλ
0
lim
ηλ − k
ω2 ∧ α = 0.
λ→ 0 X
2iπ
(2.20)
Now we have the following lemma, which will be also used in the proof of Proposition 2.2.
Lemma 2.3. The equality
lim µλ = 0
(2.21)
λ→ 0
holds.
Proof. Indeed this follows from formula (2.10), and from the fact that the class c1 (F̃)
pushes forward to 0 in H2 (X, Z) because NS(X) = 0. Then the intersection pairing
c1 F̃ , τ∗ [ω]n−1
= τ∗ c1 F̃ , [ω]n−1
X̃
X
is equal to 0, and we conclude using the fact that limλ→ 0 ωλ = τ∗ ω.
(2.22)
Then formula (2.20), and hence Proposition 2.1 follows from (2.21) and
Proposition 2.2.
Proof of Proposition 2.2. We first claim that
lim η0λ ∧ ωn−2
= 0.
λ
λ→ 0 X̃
(2.23)
Indeed, we know that the space Hdg4 (X) is perpendicular to [ω]n−2 for the intersection
1064 Claire Voisin
pairing. Hence we have
= τ∗ c2 F̃ , [ω]n−2
= 0.
c2 F̃ , τ∗ [ω]n−2
X̃
X
(2.24)
On the other hand, this is equal to
n−2
lim c2 F̃ , ωλ
λ→ 0
X̃
(2.25)
since limλ→ 0 ωλ = τ∗ ω. Exactly by the same argument, we show that
n−2
= 0.
lim c21 F̃ , ωλ
λ→ 0
X̃
(2.26)
Then the result follows from formula (2.17) and from Lemma 2.3.
Next we recall that the endomorphism R0λ of F̃, with forms coefficients is antiselfadjoint with respect to the metric kλ . This follows from the fact that Rλ is the curvature of the metric connection with respect to kλ . In a local orthonormal basis of F̃, this
will be translated into the fact that R0λ is represented by a matrix, whose coefficients are
differential 2-forms, which satisfies
t R0
λ
= −R0λ .
(2.27)
The second crucial property of R0λ is the fact that its coefficients are primitive differential
(1, 1)-forms on X̃, with respect to the metric hλ . It is well known that this implies the
following equality:
∗λ γ = −γ ∧
ωn−2
λ
,
(n − 2)!
(2.28)
where ∗λ is the Hodge ∗-operator for the metric hλ . Since hλ restricts to h on X − ∪i Vi ,
these forms satisfy as well
∗γ = −γ ∧
ωn−2
(n − 2)!
(2.29)
on X − ∪i Vi .
Now let α be a differential (2n − 4)-form on X. Then it follows from (2.29) that
there exists a positive constant cα such that for any primitive (1, 1)-form γ on X, we have
A Counterexample to the Hodge Conjecture for Kähler Varieties 1065
the following pointwise inequality of pseudo-volume forms on X:
|γ ∧ γ ∧ α| ≤ cα γ ∧ ∗γ = −cα γ ∧ γ ∧
ωn−2
.
(n − 2)!
(2.30)
Working locally in an orthonormal basis of F̃ and using the fact that the matrix R0λ is
anti-selfadjoint and with primitive coefficients of (1, 1)-type, we now get the pointwise
inequality of pseudo-volume forms on X − ∪i Vi
2
ωn−2
2
.
tr R0λ ∧ α ≤ cα tr R0λ ∧
(n − 2)!
(2.31)
Therefore, we get the inequality
X−∪i Vi
2
tr R0λ ∧ α ≤ cα
2
ωn−2
.
tr R0λ ∧
(n − 2)!
X−∪i Vi
(2.32)
But by (2.23), and because η0λ = Tr(R0λ /2iπ)2 , we know that
lim
λ→ 0 X̃
2
Tr R0λ ∧ ωn−2
= 0.
λ
(2.33)
Because the integrand is positive and ω = ωλ on X − ∪i Vi , this implies that
lim
λ→ 0 X−∪ V
i i
2
Tr R0λ ∧ ωn−2 = 0.
(2.34)
Hence
lim
λ→ 0 X−∪ V
i i
2
tr R0λ ∧ α = 0 = lim
λ→ 0 X−∪ V
i i
η0λ ∧ α.
Proposition 2.2 is proven.
(2.35)
Remark 2.4. A. Teleman mentioned to me the possibility of using the result of [1] (see the
appendix) to give a shorter proof of the equality c2 (F) = 0 for stable F. In this paper,
the results of Uhlenbeck and Yau [10] are extended to reflexive coherent stable sheaves,
and Lübke’s inequality, together with the fact that equality implies projective flatness,
is proven.
Since in our case we have a much stronger assumption than stability, namely
stability of any desingularization of F with respect to any Kähler metric, we could avoid
the reference to the technically hard result of [1] and content ourselves with an argument
which appeals only to [10] and elementary computations.
1066 Claire Voisin
3
Constructing an example
Our example will be of Weil type [12]. The Hodge classes described below have been constructed by Weil in the case of algebraic tori, as a potential counterexample to the Hodge
conjecture for algebraic varieties. In the case of a general complex torus, the construction is even simpler. These complex tori have been also considered in [13] by Zucker,
who proves there some of the results stated below. (I thank P. Deligne and C. Peters for
pointing out this reference to me.)
A complex 4-dimensional torus X with underlying lattice Γ (a lattice of rank 8),
is determined by a 4-dimensional complex subspace
W ⊂ ΓC := Γ ⊗ C
(3.1)
satisfying the condition
W ∩ ΓR = {0},
(3.2)
where ΓR = Γ ⊗ R, by the formula
X=
ΓC
.
W⊕Γ
(3.3)
We start now with a Z[I]-module structure, where I2 = −1, on such a lattice Γ ,
which makes Γ isomorphic to Z[I]4 . Then
ΓQ := Γ ⊗ Q
(3.4)
has an induced structure of K-vector space, where K is the quadratic field Q[I].
Let
ΓC = C4i ⊕ C4−i
(3.5)
be the associated decomposition into eigenspaces for I. A 4-dimensional complex torus
X with underlying lattice Γ and inheriting the I-action is determined as above by a 4dimensional complex subspace W of ΓC , which has to be the direct sum
W = Wi ⊕ W−i
of its intersections with C4i and C4−i .
(3.6)
A Counterexample to the Hodge Conjecture for Kähler Varieties 1067
We will choose W so that
dim Wi = dim W−i = 2.
(3.7)
Then W, hence X is determined by the choice of the 2-dimensional subspaces
Wi ⊂ C4i ,
W−i ⊂ C4−i ,
(3.8)
which have to be general enough so that condition (3.2) is satisfied.
We have the isomorphisms
∼ H4 (X, Q) =
∼
H4 (X, Q) =
4
ΓQ .
(3.9)
Consider the subspace
4
ΓQ ⊂
4
ΓQ .
(3.10)
K
It is defined as follows: note that we have the decomposition
ΓK := ΓQ ⊗ K = ΓK,i ⊕ ΓK,−i
(3.11)
into eigenspaces for the I action. Then (3.10) is defined as the image of
4
K ΓK,i
⊂
4
K ΓK
under the trace map
4
ΓK =
4
ΓQ ⊗ K −→
4
ΓQ .
(3.12)
Q
K
Since ΓQ is a 4-dimensional K-vector space,
4
K ΓQ
is a 1-dimensional K-vector
space, and its image is a 2-dimensional Q-vector space. We include for the convenience
of the reader a proof of the following lemma.
Lemma 3.1 (Weil [12]). The subspace
4
K ΓQ
⊂ H4 (X, Q) consists of Hodge classes, that
is, it is contained in the subspace H2,2 (X) for the Hodge decomposition.
Proof. Notice that under the isomorphisms (3.9), tensorized by C, H2,2 (X) is equal to the
image of
2
in
4
ΓC .
W⊗
2
W
(3.13)
1068 Claire Voisin
To prove the lemma, we note that we have the natural inclusion
ΓK ⊂ ΓC
(3.14)
and the equality
ΓK,i = ΓK ∩ C4i .
(3.15)
The space ΓK,i is a 4-dimensional K-vector space which generates over R the space C4i .
4
4 4
4
ΓC generates over C the line
Ci .
It follows that the image of K ΓK,i in
But we know that C4i is the direct sum of the two spaces Wi and W−i which are
2-dimensional. Hence
4
2
C4i =
is contained in
Wi ⊗
2
W⊗
2
2
W−i
(3.16)
W, that is in H2,2 (X).
To conclude the construction of an example satisfying the conclusion of
Proposition 2.1, and hence the proof of Theorem 1.1, it remains now only to prove that
a general X as above satisfies the assumptions stated at the beginning of Section 2.
4
We show first of all that the Hodge classes in K ΓQ constructed above are perpendicular to [ω]2 for a Kähler class [ω] ∈ H1,1 (X). To see this, note that with the notations
2
2
Wi ⊗
W−i , with
as above, these classes lie in
Wi ⊂ W, W−i ⊂ W.
(3.17)
∼ ΓC , the space H1,0 (X) identifies to W ∗ and
Via the natural isomorphism H1 (X, C) =
∗
∗
accordingly the space H1,1 (X) identifies to W ⊗ W ∗ . For [ω] ∈ W ⊗ W ∗ , the pairing
2
2
Wi ⊗
W−i , is obtained by squaring [ω] to get an ele[ω]2 , H2,2 (X), restricted to
ment of
2
∗
W ⊗
2
∼
W∗ =
and by projecting to
2
2
W∗ ⊗
Wi∗ ⊗
2
2
W
∗
(3.18)
∗
W−i .
Now choose
∗
∗
∗
[ω] ∈ Wi ⊗ Wi∗ ⊕ W−i ⊗ W−i
.
∗
∗
∗
(3.19)
Since W = Wi ⊕ W−i , we can find a Kähler class [ω] in this space. On the other hand,
A Counterexample to the Hodge Conjecture for Kähler Varieties 1069
we see that [ω]2 belongs to the space
2
∗
Wi ⊗
2
∗
∗
∗
Wi∗ ⊕ Wi ⊗ W−i ⊗ Wi∗ ⊗ W−i
⊕
2
∗
W−i ⊗
2
∗
W−i
.
Hence its projection (after switching the factors in the tensor product) to
2
∗
W−i is equal to 0.
(3.20)
2
Wi∗ ⊗
Concerning assumption (2), namely the fact that X does not contain any proper
positive dimensional analytic subset, we note that, since X is a complex torus, this
property will be satisfied if NS(X) = 0 and X is simple. Indeed, it is known that if Y ⊂ X is
a proper positive dimensional subvariety of a simple complex torus, then Y has positive
canonical bundle. But X being simple, Y must generate X as a group, and then X must be
algebraic, contradicting the fact that NS(X) = 0.
In conclusion, the assumptions at the beginning of Section 2 will be a consequence of the following facts.
Proposition 3.2. For a general X as above, the following hold.
(1) NS(X) = 0;
(2) X is simple;
(3) the space Hdg4 (X) is equal to the space
4
ΓQ ⊂
4
∼ H4 (X, Q).
ΓQ = H4 (X, Q) =
K
(3.21)
Proof. The analogues of these statements have been proven in the algebraic case in [12]
(see also [11]). The result is that for a general abelian 4-fold of Weil type, the NéronSeveri group is of rank 1, generated by a class ω, and the space Hdg4 (X) is of rank 3,
4
generated over Q by the space K ΓQ and by the class ω2 . Furthermore, property (2) is
true for the generic abelian variety X of Weil type.
Property (2) for the general complex torus of Weil type follows immediately,
since this is a property satisfied away from the countable union of closed analytic subsets of the moduli space of complex tori of Weil type.
As for properties (1) and (3), we prove them by an infinitesimal argument, starting from an abelian 4-fold of Weil type X satisfying the properties stated above. Assume
we can show that for some first order deformation u ∈ H1 (TX ), tangent to the moduli
space of complex tori of Weil type (which is smooth), we have
int(u)(ω) = 0 in H2 OX ,
(3.22)
1070 Claire Voisin
where the interior product here is composed of the cup product
H1 TX ⊗ H1 ΩX −→ H2 TX ⊗ ΩX
(3.23)
and of the map induced by the contraction
H2 TX ⊗ ΩX −→ H2 OX .
(3.24)
Then from the general theory of Hodge loci [4], it will follow that for a general complex
torus of Weil type, we have NS(X) = 0. Furthermore, it will also follow that
int(u) ω2 = 0 in H3 ΩX ,
(3.25)
because it is equal to 2ω ∪ int(u)(ω) and the cup product with ω from H2 (OX ) to H3 (ΩX )
is injective because ω is a Kähler class on X. But as before this will imply by the theory
of Hodge loci that for a general complex torus of Weil type we have rk Hdg4 (X) = 2 so
4
that Hdg4 (X) = K ΓQ .
Hence it remains only to find such u, which is equivalent to prove that if for
any u tangent to the moduli space of complex tori of Weil type, int(u)(ω) = 0 in H2 (OX ),
then ω = 0 in H1 (ΩX ). Here the notations are as in the beginning of this section. The
tangent space to the deformations of the complex torus X is equal to
Hom W, ΓC /W = Hom W, W = W ∗ ⊗ W.
(3.26)
The tangent space to the deformations of X as a complex torus of Weil type is then the
subspace
∗
⊗ Wi .
Hom Wi , C4i /Wi ⊕ Hom W−i , C4−i /W−i = Wi∗ ⊗ W−i ⊕ W−i
(3.27)
Via the identification (3.26), the interior product
2
W∗
H1 X, TX ⊗ H1 X, ΩX −→ H2 X, OX =
(3.28)
is equal to the contraction followed by the wedge product
∗
W ∗ ⊗ W ⊗ W ⊗ W ∗ −→
2
W∗.
(3.29)
A Counterexample to the Hodge Conjecture for Kähler Varieties 1071
We now write
ω = ω1 + ω2 + ω3 + ω4 ,
(3.30)
where
∗
ω1 ∈ Wi ⊗ Wi∗ ,
∗
∗
ω2 ∈ Wi ⊗ W−i
,
∗
(3.31)
∗
ω3 ∈ W−i ⊗ Wi∗ ,
∗
ω4 ∈ W−i ⊗ W−i
.
Then clearly for u1 ∈ Wi∗ ⊗ W−i we have
int u1 ω1 = int u1 ω2 = 0,
int u1 ω3 ∈
2
Wi∗ ,
(3.32)
int u1 ω4 ∈
Wi∗
⊗
∗
W−i
.
∗
⊗ Wi we have
Similarly, for u2 ∈ W−i
int u2 ω3 = int u1 ω4 = 0,
int u2 ω2 ∈
2
∗
W−i
,
int u2 ω1 ∈
(3.33)
Wi∗
⊗
∗
W−i
.
The condition
int u1 (ω) = 0 = int u2 (ω) = 0
(3.34)
for any u1 , u2 then implies that
2
int u1 ω3 = 0 in
Wi∗ ,
int u2 ω1 = 0 in
Wi∗
⊗
∗
W−i
,
∗
int u1 ω4 = 0 in Wi∗ ⊗ W−i
,
int u2 ω2 = 0 in
2
(3.35)
∗
W−i
,
for any u1 , u2 . But it is obvious that it implies ω1 = ω2 = ω3 = ω4 = 0.
Hence Proposition 3.2 is proven, which together with Proposition 2.1 completes
the proof of Theorem 1.1.
Appendix
Our goal in this appendix is to give a few geometric consequences of the following result,
due to Bando and Siu (the second statement given below is only a particular case of [1,
Corollary 3], namely the case where the considered sheaf has trivial rational first Chern
class).
1072 Claire Voisin
Theorem A.1. Let X be a compact Kähler variety, endowed with a Kähler metric h and
let F be a reflexive h-stable sheaf on X. Then there exists a Hermite-Einstein metric on
F relative to h. It follows that if we have c2 (F), [ω]n−2 X = 0 and c1 (F) = 0, F is locally
free and the associated metric connection is flat.
Here [ω] is the Kähler class of the metric h.
Remark A.2. Once we know that the metric connection is flat away from the singular
locus Z of F, the fact that F is locally free is immediate. Indeed, the flat connection is
associated to a local system on X − Z. But since codim Z ≥ 2, this local system extends
to X. Hence there exists a holomorphic vector bundle E on X, which admits a unitary flat
connection and is isomorphic to F away from Z. But since F is reflexive, the isomorphism
∼ F on X − Z extends to X.
E=
We assume now that X is compact Kähler and satisfies the condition that the
group NS(X) ⊗ Q = Hdg2 (X) of rational Hodge classes of degree 2 vanishes, and that the
group Hdg4 (X) is perpendicular for the intersection pairing to [ω]n−2 for some Kähler
class ω on X. Under these assumptions, X does not contain any proper analytic subset
of codimension less than or equal to 2, and any coherent sheaf F satisfies the conditions
c1 (F) = 0,
c2 (F), [ω]n−2
X
= 0.
(A.1)
We now prove the following proposition.
Proposition A.3. If X is as above, for any torsion free coherent sheaf F on X, there exist
a holomorphic vector bundle E on X, whose all rational Chern classes ci (E), i > 0 vanish,
and an exact sequence
0 −→ F −→ E −→ T −→ 0
(A.2)
where T is a torsion sheaf on X.
Before proving the proposition, we state the following corollaries.
Corollary A.4. If E is a holomorphic vector bundle on X, then all rational Chern classes
ci (E), i > 0, vanish.
Proof. Indeed, we know that there exists an inclusion
E
E ,
(A.3)
A Counterexample to the Hodge Conjecture for Kähler Varieties 1073
where E is a vector bundle of the same rank as E and satisfies the property that all
rational Chern classes ci (E ), i > 0, vanish. Now since Hdg2 (X) = 0, X does not contain
any hypersurface, and it follows that the inclusion above is an isomorphism.
Corollary A.5. If X is as above and Z ⊂ X is a nonempty proper analytic subset, the ideal
sheaf IZ does not admit a finite free resolution.
Proof. Indeed, if such a resolution
0 −→ En −→ · · · −→ Ei −→ Ei−1 −→ E0 −→ IZ −→ 0
(A.4)
would exist, then we would get the equality
c IZ = Πi c Ei i
(A.5)
with 1i = (−1)i . But the left-hand side does not vanish identically in positive degrees
since its term of degree r = codim Z is a nonzero multiple of the class of Z. On the other
hand, the right-hand side vanishes in positive degrees by Corollary A.4.
Note that the assumptions are satisfied by a general complex torus of dimension
at least 3. Taking for Z a point, we get an explicit example of a coherent sheaf which
does not admit a finite locally free resolution.
Proof of Proposition A.3. We use again induction on the rank. Let F be a torsion free
coherent sheaf of rank k on X, and assume first that F does not contain any nonzero
subsheaf of smaller rank. There is an inclusion
F
F∗∗
(A.6)
whose cokernel is a torsion sheaf, where the bidual F∗∗ of F is reflexive. Then F∗∗ does
not contain any nonzero subsheaf of smaller rank and hence is stable with respect to
the given Kähler metric h on X. The theorem of Bando and Siu [1] together with the fact
that
c1 F∗∗ = 0,
∗∗ c2 F , [ω]n−2 X = 0
(A.7)
implies that F∗∗ is a holomorphic vector bundle which is endowed with a flat connection,
hence has trivial rational Chern classes and the result is proved in this case.
Assume, otherwise, that there is an exact sequence
0 −→ G −→ F −→ H −→ 0,
(A.8)
1074 Claire Voisin
where the ranks of G and H are smaller than the rank of F, and G and H are without
torsion. This exact sequence determines (and is determined by) an extension class
e ∈ Ext1 (H, G).
(A.9)
(Here and in the following, all the extension groups refer to extensions as OX -modules.)
Now, by induction on the rank we may assume that we have inclusions
G
E1 ,
E2 ,
H
(A.10)
whose cokernels Ti are of torsion and where the Ei ’s are holomorphic vector bundles
with vanishing Chern classes. The extension class e gives first an extension class f ∈
Ext1 (H, E1 ), which provides a sheaf E containing F in such way that E /F is of torsion,
and fitting into an exact sequence
0 −→ E1 −→ E −→ H −→ 0.
(A.11)
Next, because the torsion sheaf T2 is supported in codimension ≥ 3, the restriction map
provides an isomorphism
∼ Ext1 H, E1 .
Ext1 E2 , E1 =
(A.12)
Indeed, this is implied using the long exact sequence of Ext’s by the vanishing
Ext1 T2 , E1 = Ext2 T2 , E1 = 0.
(A.13)
By the local-to-global spectral sequence for Ext’s, this in turn is implied by the vanishing
Exti T2 , E1 = 0,
i ≤ 2.
(A.14)
Since E1 is locally free, and the statement is local, this follows now from the vanishing
Exti T, OX = 0,
i ≤ j,
(A.15)
for any torsion sheaf T supported in codimension ≥ j + 1.
The surjectivity of (A.12) implies that f is the image under the restriction map of
a class g ∈ Ext1 (E2 , E1 ). Hence it follows that there is a holomorphic vector bundle E on
X, which is an extension of E2 by E1 , and which contains E as a subsheaf, such that the
quotient E/E is of torsion. The vector bundle E has vanishing rational Chern classes,
because Ei satisfy this property for i = 1, 2, and contains F as a subsheaf such that the
quotient E/F is of torsion. This completes the proof by induction.
A Counterexample to the Hodge Conjecture for Kähler Varieties 1075
Acknowledgments
I would like to thank Joseph Le Potier and Andrei Teleman for helpful discussions on this paper. The
references [2, 9] have been a starting point for this work. I also thank Pierre Deligne and Luc Illusie
for their interest and their questions, and the referee for many constructive comments.
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Claire Voisin: Institut de Mathématiques de Jussieu, CNRS, Unité Mixte de Recherche 7586, France
E-mail address: [email protected]